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$a)$ In the figure, $ABCD$ is a square. $E$ and $F$ are the midmpoint of the sides $[AD]$ and $[CD]$ respectively. The are of the shaded region is $6cm^2$. What is the length of one side of the square. enter image description here

My attemp is:$S_{\Delta ABG}=\frac{AB\cdot GG_1}{2},$ where $GG_1\perp AB$

$\Rightarrow$ $S_{\Delta ABG}=\frac{AB\cdot BC}{6},$ becouse i soppose that $GG_1=\frac{BC}{3},$ but I didint now it is correct. Now we have: $AB=6$

Please help me. Thaky very much.

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Consider the image:

enter image description here

$$A = \dfrac{3x^2}{2} = 6$$

So $x = 2$

Therefore side length $S = 3x = 6$

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From $G$ drop perpendicular on $AB$. Name it $D$.

As $\angle GDB=45^\circ$ , $GD=AD$. $$\therefore {GD\over AB-AD}={GD\over AB-GD}={EA\over AB}={1\over 2}\\\implies GD={AB\over3}$$ Now given $${1\over 2}AB\times GD=6\implies AB=6$$

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